Let's examine the given function:
[tex]\[
y = \frac{7}{3} \cot \left[\frac{1}{3} \left(x - \frac{\pi}{4}\right)\right]
\][/tex]
### (a) Period
The general form of the cotangent function is [tex]\(y = A \cot(Bx - C)\)[/tex], where [tex]\(A\)[/tex] is the amplitude, [tex]\(B\)[/tex] is the coefficient of [tex]\(x\)[/tex] that affects the period, and [tex]\(C\)[/tex] is the phase shift. The period of the cotangent function is given by:
[tex]\[
\text{Period} = \frac{\pi}{B}
\][/tex]
For the given function:
[tex]\[
B = \frac{1}{3}
\][/tex]
Substituting [tex]\(B\)[/tex] into the period formula:
[tex]\[
\text{Period} = \frac{\pi}{\frac{1}{3}} = 3\pi
\][/tex]
### (b) Phase Shift
The phase shift of a function in the form [tex]\(y = A \cot(Bx - C)\)[/tex] is given by:
[tex]\[
\text{Phase Shift} = \frac{C}{B}
\][/tex]
In the given function, [tex]\(C = \frac{\pi}{4}\)[/tex] and [tex]\(B = \frac{1}{3}\)[/tex]. Substituting these values into the formula for phase shift:
[tex]\[
\text{Phase Shift} = \frac{\frac{\pi}{4}}{\frac{1}{3}} = \frac{\pi}{4} \cdot 3 = \frac{3\pi}{4}
\][/tex]
### (c) Range
The range of the cotangent function, [tex]\(\cot(x)\)[/tex], is all real numbers, since the cotangent can take any real value as [tex]\(x\)[/tex] varies over its domain (except at its asymptotes).
To summarize:
- (a) Period: [tex]\(\boxed{3 \pi}\)[/tex]
- (b) Phase Shift: [tex]\(\boxed{\frac{3 \pi}{4}}\)[/tex]
- (c) Range: [tex]\(\boxed{\text{all real numbers}}\)[/tex]
These results match the detailed analysis.
